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354 XI ELEMENTARY SPECTRAL THEORY

(f ) For X = l_n , a necessary and sufficient condition for the Sturm-Liouville
problem to have a solution is that \ f(t)cp_n(t) dt = 0. Then, for any solution w,
c_n = (w | q>_n) is arbitrary, and for m ^ n, c_m is given by the same formula as
in (e).

The homogeneous Sturm-Liouville problem cannot have two linearly
independent solutions, otherwise it would have solutions y for which y(d)
and y'(d) are arbitrary, which is absurd; this proves (b). The fact that all
eigenvalues A_w are real follows from (1 1 .7.7) and (1 1 .5.7) ; moreover it follows
from (11 .7.4) that at most finitely many l_n are negative. By Mercer's theorem
((11.6.7) and (11.6.8)), we have for the Green function

(11.7.10.1) K(*,x) = £|<p_n(0<P,,(x)

n A_n

the series being absolutely and uniformly convergent in I x I (it is supposed,
as we may, that 0 is not one of the !„). We observe that (d) follows from
(11.6.3) and (11.7.8) when the additional assumption is made on w that M/
is continuous in I. To prove (d) in general, let t_i (1 < i < m) be the points of
1 where w^f has a discontinuity, and let a,- = w'(^-i-) — w'(fj— ). Then the

function v = w + £ ^aiK_f. satisfies all the conditions of (d) and in addition

i=l

has a continuous derivative, by (11.7.6). Using (11.7.10.1) we conclude the
proof of (d). From the fact that the identity mapping of E = ^c(I) ^^nto Gr is
continuous, it follows that for the functions w satisfying the conditions of
(d), we can also write w = £ c_n q>_n , the sequence being convergent in the

prehilbert space G. To prove (c) it will then be enough to show that the set P
of these functions w is dense in G. Now, for any function M e G, consider the
continuous function w_m equal to u in

to a linear function x -» ax + /? satisfying the first (resp. second) boundary
condition (1 1 .7.2) in

and to a linear function in each of the intervals

+ -L,_fl+il, L_!,6_-
2m m L ^ 2m



	354 XI ELEMENTARY SPECTRAL THEORY (f ) For X = l_n , a necessary and sufficient condition for the Sturm-Liouville problem to have a solution is that \ f(t)cp_n(t) dt = 0. Then, for any solution w, c_n = (w \| q>_n) is arbitrary, and for m ^ n, c_m is given by the same formula as in (e). The homogeneous Sturm-Liouville problem cannot have two linearly independent solutions, otherwise it would have solutions y for which y(d) and y'(d) are arbitrary, which is absurd; this proves (b). The fact that all eigenvalues A_w are real follows from (1 1 .7.7) and (1 1 .5.7) ; moreover it follows from (11 .7.4) that at most finitely many l_n are negative. By Mercer's theorem ((11.6.7) and (11.6.8)), we have for the Green function (11.7.10.1) K(,x) = £\|<p_n(0<P,,(x) n A_n* the series being absolutely and uniformly convergent in I x I (it is supposed, as we may, that 0 is not one of the !„). We observe that (d) follows from (11.6.3) and (11.7.8) when the additional assumption is made on w that M/ is continuous in I. To prove (d) in general, let t_i (1 < i < m) be the points of 1 where w^f has a discontinuity, and let a,- = w'(^-i-) — w'(fj— ). Then the m function v = w + £ ^aiK_f. satisfies all the conditions of (d) and in addition i=l has a continuous derivative, by (11.7.6). Using (11.7.10.1) we conclude the proof of (d). From the fact that the identity mapping of E = ^c(I) ^^nto Gr is continuous, it follows that for the functions w satisfying the conditions of (d), we can also write w = £ c_n q>_n , the sequence being convergent in the n prehilbert space G. To prove (c) it will then be enough to show that the set P of these functions w is dense in G. Now, for any function M e G, consider the continuous function w_m equal to u in

	to a linear function x -» ax + /? satisfying the first (resp. second) boundary condition (1 1 .7.2) in

	and to a linear function in each of the intervals + -L,_fl+il, L_!,6_- 2m m L ^ 2m